Bingham distribution

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Short description: Antipodally symmetric probability distribution on the n-sphere

In statistics, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric probability distribution on the n-sphere.[1] It is a generalization of the Watson distribution and a special case of the Kent and Fisher–Bingham distributions.

The Bingham distribution is widely used in paleomagnetic data analysis,[2] and has been used in the field of computer vision.[3][4][5]

Its probability density function is given by

[math]\displaystyle{ f(\mathbf{x}\,;\,M,Z) \; dS^{n-1} = {}_1 F_1 \left( \tfrac12 ; \tfrac n2 ; Z \right)^{-1} \cdot \exp \left( \operatorname{tr} Z M^T \mathbf{x} \mathbf{x}^T M \right)\; dS^{n-1} }[/math]

which may also be written

[math]\displaystyle{ f(\mathbf{x}\,;\,M,Z)\; dS^{n-1} \;=\; {}_1 F_1 \left( \tfrac12 ; \tfrac n2 ;Z \right)^{-1} \cdot \exp\left( \mathbf{x}^T M Z M^T \mathbf{x} \right)\; dS^{n-1} }[/math]

where x is an axis (i.e., a unit vector), M is an orthogonal orientation matrix, Z is a diagonal concentration matrix, and [math]\displaystyle{ {}_{1}F_{1}(\cdot;\cdot,\cdot) }[/math] is a confluent hypergeometric function of matrix argument. The matrices M and Z are the result of diagonalizing the positive-definite covariance matrix of the Gaussian distribution that underlies the Bingham distribution.

See also